Brownian motion of water molecules provides an essential length scale, the diffusion length, commensurate with cell dimensions in biological tissues. Measuring the diffusion coefficient as a function of diffusion time makes in vivo diffusion MRI uniquely sensitive to the cellular features about three orders of magnitude below imaging resolution. However, there is a longstanding debate, regarding which contribution - intra- or extra-cellular - is more relevant in the overall time-dependence of the MRI-derived diffusion metrics. Here we resolve this debate in the human brain white matter. By varying not just the diffusion time, but also the gradient pulse duration of a standard diffusion MRI sequence, we identify a functional form of the measured time-dependent diffusion coefficient transverse to white matter tracts in 10 healthy volunteers. This specific functional form is shown to originate from the extra-axonal space, and provides estimates of the fiber packing correlation length for axons in a bundle. Our results offer a metric for the outer axonal diameter, a promising candidate marker for demyelination in neurodegenerative diseases. From the methodological perspective, our analysis demonstrates how competing models, which describe different physics yet interpolate standard measurements equally well, can be distinguished based on their prediction for an independent "orthogonal" measurement.